Show that \(lim_{x\rightarrow\infty}{1^2+2^2+....+(3n)^2\over (1+2+...+5n)(2n+3)}={9\over25}\)

Evaluate : \(lim_{x \rightarrow 0}{3^x-1\over \sqrt{1+x}-1}.\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{\sqrt{x^2+a^2}-a\over \sqrt{x^2+b^2}-b}\)

Evaluate the following limits :\(limx_{x\rightarrow \infty}x[{3^{1\over x}+1-cos({1\over x}) -e^{1\over x}}]\)

Describe the interval(s) on which each function is continuous.
\(h(x)= \begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{cases}\)

A tomato wholesaler finds that the price of a newly harvested tomatoes is Rs. 0.16 per kg if he purchases fewer than 100 kgs each day. However, if he purchases at least 100 kgs daily, the price drops to Rs. 0.14 per kg. Find the total cost function and discuss the cost when the purchase is 100 kgs.

A function f is defined as follows :
\(f(x)= \begin{cases}0 & \text { for } \quad x<0 \\ x & \text { for } \quad 0 \leq x<1 \\ -x^{2}+4 x-2 & \text { for } \quad 1 \leq x<3 \\ 4-x & \text { for } \quad x \geq 3\end{cases}\)
Is the function continuous?

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R
\(f(x)={x^2-2x-8\over x+2},x_o=-2\)

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R
\(f(x)={3-\sqrt{x}\over 9-x},x_o=9\)

Find the constant b that makes g continuous on \((-\infty,\infty)\)
\(g(x)= \begin{cases}x^{2}-b^{2} & \text { if } x<4 \\ b x+20 & \text { if } x \geq 4\end{cases}\)